The exercise you've provided involves a bounded sequence and a function. Let's break it down step by step:

1. Problem Setup:

• You are given a sequence $(u_n)$, where $u_0 = 0$, and for all $n \in \mathbb{N}$, the recurrence relation is defined as: $u_{n+1} = \frac{u_n + 3}{4u_n + 4}$

• There is also a function $f$ defined on the interval $]-1, +\infty[$ by: $f(x) = \frac{x + 3}{4x + 4}$

2. Tasks:

1. Étudier les variations de $f$:
   You need to study the variations (monotonicity) of the function $f(x)$, which means you have to find the derivative $f'(x)$ and determine where the function is increasing or decreasing.

2. Démontrer par récurrence que pour tout entier naturel $n$, $0 \leq u_n \leq 1$:
   This asks you to prove by mathematical induction that for all natural numbers $n$, the sequence $(u_n)$ stays within the bounds $0 \leq u_n \leq 1$.

Solution Outline:

1. Study the Variations of $f(x)$

The function is: $f(x) = \frac{x + 3}{4x + 4}$ To study the variations, we compute its derivative:

• First, apply the quotient rule to $f(x) = \frac{x + 3}{4x + 4}$.

The quotient rule is:
$f'(x) = \frac{(4x + 4) \cdot 1 - (x + 3) \cdot 4}{(4x + 4)^2}$ Simplifying the numerator and denominator will give us the derivative $f'(x)$.

2. Prove by Induction that $0 \leq u_n \leq 1$

To prove by induction:

• Base Case: Verify for $n = 0$.
  We know that $u_0 = 0$, and $0 \leq 0 \leq 1$, so the base case holds.

• Inductive Step:
  Assume that $0 \leq u_n \leq 1$ holds for some $n \geq 0$. You need to show that this implies $0 \leq u_{n+1} \leq 1$.

  Using the recurrence relation: $u_{n+1} = \frac{u_n + 3}{4u_n + 4}$ We can analyze the behavior of this function within the bounds of $0 \leq u_n \leq 1$ to prove the step.

Would you like me to proceed with solving both steps in detail? Let me know if you'd like further clarification!

Here are 5 related questions that might interest you:

1. How do we determine the critical points of a function using its derivative?
2. What does the monotonicity of a function tell us about its behavior?
3. How does the recurrence relation help in analyzing sequences?
4. Can the fixed points of a function be used to analyze the long-term behavior of a sequence?
5. What role does mathematical induction play in proving properties of sequences?

Tip: When applying the quotient rule, remember to carefully simplify the numerator to avoid errors in sign or expression.